Secant varieties of toric varieties

Let $X_P$ be a smooth projective toric variety of dimension $n$ embedded in \PP^r using all of the lattice points of the polytope $P$. We compute the dimension and degree of the secant variety \Sec X_P. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties $X_A$ embedded using a set of lattice points A \subset P\cap\ZZ^n containing the vertices of $P$ and their nearest neighbors.Comment: v1, AMS LaTex, 5 figures, 25 pages; v2, reference added; v3, This is a major rewrite. We have strengthened our main results to include a classification of smooth lattice polytopes P such that Sec X_P does not have the expected dimension. (See Theorems 1.4 and 1.5.) There was also a considerable amount of reorganization, and some expository material was eliminated; v4, 28 pages, minor corrections, additional and updated reference

Liouville Black Holes

The dynamics of Liouville fields coupled to gravity are investigated by applying the principle of general covariance to the Liouville action in the context of a particular form of two-dimensional dilaton gravity. The resultant field equations form a closed system for the Liouville/gravity interaction. A large class of asymptotically flat solutions to the field equations is obtained, many of which can be interpreted as black hole solutions. The temperature of such black holes is proportional to their mass-parameters. An exact solution to the back reaction problem is obtained to one-loop order, both for conformally coupled matter fields and for the quantized metric/Liouville system. Quantum effects are shown to map the space of classical solutions into one another. A scenario for the end-point of black-hole radiation is discussed.Comment: 32 pgs., WATPHYS-TH93/03 (Latex plus two postscript figures appended

Incorporating chemical signalling factors into cell-based models of growing epithelial tissues

In this paper we present a comprehensive computational framework within which the effects of chemical signalling factors on growing epithelial tissues can be studied. The method incorporates a vertex-based cell model, in conjunction with a solver for the governing chemical equations. The vertex model provides a natural mesh for the finite element method (FEM), with node movements determined by force laws. The arbitrary Lagrangian–Eulerian formulation is adopted to account for domain movement between iterations. The effects of cell proliferation and junctional rearrangements on the mesh are also examined. By implementing refinements of the mesh we show that the finite element (FE) approximation converges towards an accurate numerical solution. The potential utility of the system is demonstrated in the context of Decapentaplegic (Dpp), a morphogen which plays a crucial role in development of the Drosophila imaginal wing disc. Despite the presence of a Dpp gradient, growth is uniform across the wing disc. We make the growth rate of cells dependent on Dpp concentration and show that the number of proliferation events increases in regions of high concentration. This allows hypotheses regarding mechanisms of growth control to be rigorously tested. The method we describe may be adapted to a range of potential application areas, and to other cell-based models with designated node movements, to accurately probe the role of morphogens in epithelial tissues

Classification and Identification of Pfiesteria and Pfiesteria-Like Species

Dinoflagellates can be classified both botanically and zoologically; however, they are typically put in the botanical division Pyrrhophyta. As a group they appear most related to the protistan ciliates and apicomplexans at the ultrastructure level. Within the Pyrrhophyta are both unarmored and armored forms of the dominant, motile flagellated stage. Unarmored dinoflagellates do not have thecal or wall plates arranged in specific series, whereas armored species have plates that vary in thickness but are specific in number and arrangement. In armored dinoflagellates, the plate pattern and tabulation is a diagnostic character at the family, subfamily, and even genus levels. In most cases, the molecular characterization of dinoflagellates confirms the taxonomy on the basis of external morphology; this has been demonstrated for several groups. Together, both genetic and morphological criteria are becoming increasingly important for the characterization, separation, and identification of dinoflagellates species. Pfiesteria and Pfiesteria-like species are thinly armored forms with motile dinospore stages characterized by their distinct plate formulae. Pfiesteria piscicida is the best-known member of the genus; however, there is at least one other species. Other genetically and morphologically related genera, now grouped under the common names of Lucy, Shepherd\u27s crook, and cryptoperidiniopsoid, are being studied and described in separate works. All these other heterotrophic dinoflagellate groups, many of which are thought to be benign, co-occur in estuarine waters where Pfiesteria has been found